where \(G_{MN}\)  denotes the metric in ten dimensions and \(g_{\mu \nu }\)  the metric on the nine-dimensional worldvolume of the brane. Moreover, one has \(\kappa _{10}^2=\ell _s^8/(4\pi )\) , with \(l_s\)  the string length.
Note that this is actually the same action as considered in [1]}.
The only slight difference is the presence of a single source instead of two. Indeed, the second source would sit at \(r=R\)  and it is here taken into account by extending periodically the solution beyond \(r \in [-\frac{R}{2}, \frac{R}{2}]\) .
